Second order differential operators and Dirichlet integrals with singular coefficients. I: Functional calculus of one-dimensional operators. (English) Zbl 0653.35034

Gaveau, Bernard; Okada, Masami; Okada, Tatsuya

Tôhoku Math. J., II. Ser. 39, 465-504 (1987).

Let a,c be piecewise \(C^ 1\), \(C^ 0\) functions on \({\mathbb{R}}\) respectively. The authors study the parabolic Cauchy problem \[ \partial u/\partial t=Lu\quad if\quad t>0,\quad u|_{t=0}=0, \] where \(Lu=(1/c^ 2)\partial_ x(1/a\) \(2)\partial_ xu)\) (and also Cauchy problems for wave or Schrödinger equations). Explicit formulas for the heat kernel are given. The case where L is a spherically symmetric elliptic operator in \({\mathbb{R}}^ 3\) is also treated.

Reviewer: P.Godin

MSC:

35K15	Initial value problems for second-order parabolic equations
35L15	Initial value problems for second-order hyperbolic equations
35R05	PDEs with low regular coefficients and/or low regular data
35C05	Solutions to PDEs in closed form

Keywords:

singular coefficients; parabolic Cauchy problem; Schrödinger equations; Explicit formulas; heat kernel; spherically symmetric